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Shocks in asymmetric simple exclusion processes of interacting particles
Abstract
In this paper, we study shocks and related transitions in asymmetric simple exclusion processes of particles with nearest-neighbor interactions. We consider two kinds of interparticle interactions. In one case, the particle-hole symmetry is broken due to the interaction. In the other case, particles have an effective repulsion due to which the particle current density drops down near one-half filling. These interacting particles move on a one-dimensional lattice which is open at both the ends with injection of particles at one end and withdrawal of particles at the other. In addition to this, there are possibilities of attachments or detachments of particles to or from the lattice with certain rates. The hydrodynamic equation that involves the exact particle current density of the particle conserving system and additional terms taking care of the attachment-detachment kinetics is studied using the techniques of boundary layer analysis.
... In [21], the authors showed by computer simulations and mean-field arguments that, in this nonconserved dynamics, one can have phase coexistence where low and high density phases are separated by stable discontinuities (domain wall) in the density profile. Recently, this dynamics was also studied theoretically in [18] [19]. ...
In this research, the totally asymmetric exclusion process without particle
number conservation is discussed. Based on the mean field approximation and the
Rankine-Hugoniot condition, the necessary and sufficient conditions of the
existence of the domain wall have been obtained. Moreover, the properties of
the domain wall, including the location and height, have been studied
theoretically. All the theoretical results are demonstrated by the computer
simulations.
We study a driven many particle system comprising of two identical lanes of
finite lengths. On one lane, particles hop diffusively with a bias in a
specific direction. On the other lane, particles hop in a specific direction
obeying mutual exclusion rule. In addition, the two lanes are connected with
each other through exchange of particles with certain rules. The system, at its
two ends, is in contact with particle reservoirs which maintain specific
particle densities at the two ends. In this paper, we study boundary-induced
phase transitions exhibited by this system and predict the phase diagram using
the technique of fixed point based boundary layer analysis. An interesting
manifestation of the interplay of two density variables associated with two
lanes is found in the shock phase in which the particle density profile across
the lane with unidirectional hopping shows a jump discontinuity (shock) from a
low- to a high-density region. The density profile on the diffusion-lane never
exhibits a shock. However, the shock in the other lane gives rise to a
discontinuity in the slope of the diffusion-lane density profile. We show how
an approximate solution for the slope can be obtained in the boundary layer
analysis framework.
This Letter investigates a totally asymmetric simple exclusion process (TASEP) with junctions in a one-dimensional transport system. Parallel update rules and periodic boundary condition are adopted. Two cases corresponding to different update rules are studied. The results show that the stationary states of system mainly depend on the selection behavior of particle at the bifurcation point.Highlights► For no preference case, the system exists three stationary phases. ► For preference case, the system exists five stationary phases. ► The road lengths have not qualitative influence on the fundamental diagram.
Motor proteins in living cells, such as kinesin and dynein, can move processively along the microtubule (MT), and can also detach from or attach to MT stochastically. Experiments found that the traffic of motors along MT may be jammed; thus various theoretical models were designed to understand this process. However, previous studies mainly focused on motor attachment/detachment rate dependent properties. Leduc et al. recently found that the traffic jam of motor protein Kip3 depends on MT length (Proc. Natl. Acad. Sci. U.S.A. 109, 6100 (2012)). Therefore, this study discusses the MT length-dependent properties of motor traffic. The results showed that MT length has one critical value N
c
; a traffic jam occurs only when MT length N > N
c
. The jammed MT length increases with total MT length N , whereas the non-jammed MT length may not change monotonically with N . The critical value Nc increases with motor detachment rate from MT, but decreases with motor attachment rate to MT. Therefore, the traffic of motors will be more likely to be jammed when the MT is long, motor detachment rate is high, and motor detachment rate is low.
Graphical abstract
Boundary-induced phase transitions in a driven diffusive process can be studied through a phase-plane analysis of the boundary-layer equations. In this paper, we generalize this approach further to show how various shapes including multishocks and downward shocks in the bulk particle density profile can be understood by studying the dependence of the fixed points of the boundary-layer equation on an appropriate parameter. This is done for a particular driven interacting particle system as a prototypical example. The present analysis shows the special role of a specific bifurcation of the fixed points in giving rise to different kinds of shocks.
We extend one dimensional asymmetric simple exclusion process (ASEP) to a surface and show that the effect of transverse diffusion is to induce a continuous phase transition from a constant density phase to a maximal current phase as the forward transition probability p is tuned. The signature of the nonequilibrium transition is in the finite size effects near it. The results are compared with similar couplings operative only at the boundary. It is argued that the nature of the phases can be interpreted in terms of the modifications of boundary layers.
We show the existence of a nonequilibrium tricritical point induced by a
repulsive interaction in one dimensional asymmetric exclusion process. The
tricritical point is associated with the particle-hole symmetry breaking
introduced by the repulsion. The phase diagram and the crossover in the
neighbourhood of the tricritical point for the shock formation at one of the
boundaries are determined.
The steady-state shock formation in processes such as nonconserving asymmetric simple exclusion processes in varied situations is shown to have a precursor of a critical deconfinement transition on the low-density side. The diverging length scales and the quantitative description of the transition are obtained from a few general properties of the dynamics without relying on specific details.
Several recent works have shown that the one-dimensional fully asymmetric exclusion model, which describes a system of particles hopping in a preferred direction with hard core interactions, can be solved exactly in the case of open boundaries. Here the authors present a new approach based on representing the weights of each configuration in the steady state as a product of noncommuting matrices. With this approach the whole solution of the problem is reduced to finding two matrices and two vectors which satisfy very simple algebraic rules. They obtain several explicit forms for these non-commuting matrices which are, in the general case, infinite-dimensional. Their approach allows exact expressions to be derived for the current and density profiles. Finally they discuss briefly two possible generalizations of their results: the problem of partially asymmetric exclusion and the case of a mixture of two kinds of particles.
We consider the asymmetric simple exclusion process (ASEP) with open boundaries and other driven stochastic lattice gases of particles entering, hopping and leaving a one-dimensional lattice. The long-term system dynamics, stationary states, and the nature of phase transitions between steady states can be understood in terms of the interplay of two characteristic velocities, the collective velocity and the shock (domain wall) velocity. This interplay results in two distinct types of domain walls whose dynamics is computed. We conclude that the phase diagram of the ASEP is generic for one-component driven lattice gases with a single maximum in the current-density relation.
The traffic of molecular motors through open tube-like compartments is studied using lattice models. These models exhibit boundary-induced phase transitions related to those of the asymmetric simple exclusion process (ASEP) in one dimension. The location of the transition lines depends on the boundary conditions at the two ends of the tubes. Three types of boundary conditions are studied: (A) Periodic boundary conditions which correspond to a closed torus-like tube. (B) Fixed motor densities at the two tube ends where radial equilibrium holds locally; and (C) Diffusive motor injection at one end and diffusive motor extraction at the other end. In addition to the phase diagrams, we also determine the profiles for the bound and unbound motor densities using mean field approximations and Monte Carlo simulations. Our theoretical predictions are accessible to experiments.
We study the formation of localized shocks in one-dimensional driven diffusive systems with spatially homogeneous creation and annihilation of particles (Langmuir kinetics). We show how to obtain hydrodynamic equations that describe the density profile in systems with uncorrelated steady state as well as in those exhibiting correlations. As a special example of the latter case, the Katz-Lebowitz-Spohn model is considered. The existence of a localized double density shock is demonstrated in one-dimensional driven diffusive systems. This corresponds to phase separation into regimes of three distinct densities, separated by localized domain walls. Our analytical approach is supported by Monte Carlo simulations.
We investigate boundary-driven phase transitions in open driven diffusive systems. The generic phase diagram for systems with short-ranged interactions is governed by a simple extremal principle for the macroscopic current, which results from an interplay of density fluctuations with the motion of shocks. In systems with more than one extremum in the current-density relation, one finds a minimal current phase even though the boundaries support a higher current. The boundary layers of the critical minimal current and maximal current phases are argued to be of a universal form. The predictions of the theory are confirmed by Monte Carlo simulations of the two-parameter family of stochastic particle hopping models of Katz, Lebowitz, and Spohn and by analytical results for a related cellular automaton with deterministic bulk dynamics. The effect of disorder in the particle jump rates on the boundary layer profile is also discussed.
We study a one-dimensional totally asymmetric exclusion process with random particle attachments and detachments in the bulk. The resulting dynamics leads to unexpected stationary regimes for large but finite systems. Such regimes are characterized by a phase coexistence of low and high density regions separated by domain walls. We use a mean-field approach to interpret the numerical results obtained by Monte Carlo simulations, and we predict the phase diagram of this nonconserved dynamics in the thermodynamic limit.
We investigate shock formation in an asymmetric exclusion process with creation and annihilation of particles in the bulk. We show how the continuum mean-field equations can be studied analytically and hence derive the phase diagrams of the model. In the large system-size limit direct simulations of the model show that the stationary state is correctly described by the mean-field equations, thus the predicted mean-field phase diagrams are expected to be exact. The emergence of shocks and the structure of the phase diagram are discussed. We also analyze the fluctuations of the shock position by using a phenomenological random walk picture of the shock dynamics. The stationary distribution of shock positions is calculated, by virtue of which the numerically determined finite-size scaling behavior of the shock width is explained.
We discuss a class of driven lattice gas obtained by coupling the one-dimensional totally asymmetric simple exclusion process to Langmuir kinetics. In the limit where these dynamics are competing, the resulting nonconserved flow of particles on the lattice leads to stationary regimes for large but finite systems. We observe unexpected properties such as localized boundaries (domain walls) that separate coexisting regions of low and high density of particles (phase coexistence). A rich phase diagram, with high and low density phases, two and three phase coexistence regions, and a boundary independent "Meissner" phase is found. We rationalize the average density and current profiles obtained from simulations within a mean-field approach in the continuum limit. The ensuing analytic solution is expressed in terms of Lambert W functions. It allows one to fully describe the phase diagram and extract unusual mean-field exponents that characterize critical properties of the domain wall. Based on the same approach, we provide an explanation of the localization phenomenon. Finally, we elucidate phenomena that go beyond mean-field such as the scaling properties of the domain wall.
We study the formation of localized shocks in one-dimensional driven diffusive systems with spatially homogeneous creation and annihilation of particles (Langmuir kinetics). We show how to obtain hydrodynamic equations that describe the density profile in systems with uncorrelated steady state as well as in those exhibiting correlations. As a special example of the latter case, the Katz-Lebowitz-Spohn model is considered. The existence of a localized double density shock is demonstrated in one-dimensional driven diffusive systems. This corresponds to phase separation into regimes of three distinct densities, separated by localized domain walls. Our analytical approach is supported by Monte Carlo simulations.
Systems with two species of active molecular motors moving on (cytoskeletal) filaments into opposite directions are studied theoretically using driven lattice gas models. The motors can unbind from and rebind to the filaments. Two motors are more likely to bind on adjacent filament sites if they belong to the same species. These systems exhibit (i) Continuous phase transitions towards states with spontaneously broken symmetry, where one motor species is largely excluded from the filament, (ii) Hysteresis of the total current upon varying the relative concentrations of the two motor species, and (iii) Coexistence of traffic lanes with opposite directionality in multi-filament systems. These theoretical predictions should be experimentally accessible.
We consider an exclusion process, with particles injected with rate at the origin and removed with rate at the right boundary of a one-dimensional chain of sites. The particles are allowed to hop onto unoccupied sites, to the right only. For the special case of the model was solved previously by Derrida et al. Here we extend the solution to general . The phase diagram obtained from our exact solution differs from the one predicted by the mean field approximation. Comment: LATEX (23 pages, one figure only partially included, complete figure available upon request), Weizmann preprint WIS-92/96/Dec.-PH
The idea of duality in one-dimensional nonequilibrium transport is introduced by generalizing the observations by S. Mukherji and V. Mishra [Phys. Rev. E (3) 74, No. 1, 011116, 11 pp. (2006)]. A general approach is developed for the classification and characterization of the steady state phase diagrams which are shown to be determined by the nature of the zeros of a set of coarse-grained functions that encode the microscopic dynamics. A new class of nonequilibrium multicritical points has been identified.
We investigate theoretically and via computer simulation the stationary nonequilibrium states of a stochastic lattice gas under the influence of a uniform external fieldE. The effect of the field is to bias jumps in the field direction and thus produce a current carrying steady state. Simulations on a periodic 30 30 square lattice with attractive nearest-neighbor interactions suggest a nonequilibrium phase transition from a disordered phase to an ordered one, similar to the para-to-ferromagnetic transition in equilibriumE=0. At low temperatures and largeE the system segregates into two phases with an interface oriented parallel to the field. The critical temperature is larger than the equilibrium Onsager value atE=0 and increases with the field. For repulsive interactions the usual equilibrium phase transition (ordering on sublattices) is suppressed. We report on conductivity, bulk diffusivity, structure function, etc. in the steady state over a wide range of temperature and electric field. We also present rigorous proofs of the Kubo formula for bulk diffusivity and electrical conductivity and show the positivity of the entropy production for a general class of stochastic lattice gases in a uniform electric field.
Steady states of driven lattice gases with open boundaries are investigated. Particles are fed into the system at one edge, travel under the action of an external field, and leave the system at the opposite edge. Two types of phase transitions involving nonanalytic changes in the density profiles and the particle number fluctuation spectra are encountered upon varying the feeding rate and the particle interactions, and associated diverging length scales are identified. The principle governing the transitions is the tendency of the system to maximize the transported current.
In this paper, we study the boundary-induced phase transitions in a particle nonconserving asymmetric simple exclusion process with open boundaries. Using a boundary layer analysis on the mean field version of the model, we show that the key signatures of various bulk phase transitions are present in the boundary layers of the density profiles. In addition, we also find surface transitions in the low- and high-density phases. The surface transition in the low-density phase provides a complete description of the nonequilibrium critical point found in this system.
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